MHB Proof of the Division Algorithm

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The discussion centers on the application of the well ordering principle (WOP) to subsets of non-negative integers in the context of the division algorithm. It confirms that since any subset of non-negative integers also adheres to the WOP, it is valid to apply the principle in this scenario. The conversation touches on the clarity of the concept, with one participant questioning if their understanding was overly pedantic. Ultimately, the dialogue reflects a light-hearted acknowledgment of occasional confusion regarding mathematical principles. The participants agree on the correctness of applying WOP to non-negative integers.
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In many books on number theory they define the well ordering principle (WOP) as:
Every non- empty subset of positive integers has a least element.
Then they use this in the proof of the division algorithm by constructing non-negative integers and applying WOP to this construction. Is it possible to apply the WOP to a subset of non-negative integers? Am I being too pedantic?
 
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It's rather obvious isn't it? "Applying the WOP to a subset of non-negative integers" would simply mean that, given X, a subset of the non-negative integers, any subset of X has a least member. And that is true because any subset of X is also a subset of the non-negative integers.

If that is not what you mean then please explain what you mean by "apply the WOP to a subset of non-negative integers".​

 
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Yes of course. I just had a senior moment.
Thanks.
 
There are those of use who live in "senior moments"! We are called "seniors".
 
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